The Bridge of Memories
After the screen fades to black at the Suga Shrine, Taki and Mitsuha don't just walk away. They stand on those stairs for a long time, crying and laughing. Even though their specific memories of the body-switching are gone, the feelings remain.
Taki realizes that the "hollow hole" in his heart that he felt for eight years has finally been filled. Mitsuha feels a sense of peace she hasn't known since the night of the comet.
Reconnecting with Friends
Taki introduces Mitsuha to his world in Tokyo. We learn what happened to the others:
Tessie and Sayaka: Mitsuha's best friends from Itomori moved to Tokyo together. Taki actually saw them in a cafe earlier in the story without realizing who they were! They are now engaged to be married.
Mitsuha's Family: Her sister, Yotsuha, is now a high school student in Tokyo, and her grandmother, Hitoha, lived long enough to see the "Miracle of Itomori" save her family line.
The Red Braided Cord
Mitsuha still wears the red braided cord (kumihimo) in her hair. When Taki sees it, he feels a strange jolt of electricity. He remembers holding that cord on a train years ago. It is the physical Musubi (the bond) that survived time, space, and even death to bring them back together.
A Future Together
Director Makoto Shinkai has suggested in interviews and his later film Weathering With You (where Taki and Mitsuha make cameo appearances) that:
They get married: In a small Easter egg, a photo in a later movie shows Taki and Mitsuha's wedding.
The Memories Return: While they never remember every detail of the switching, they spend their lives sharing stories of their "dreams," slowly piecing together the miracle that happened when they were teenagers.